Download Evolution of Primordial Magnetic Fields from Phase

NORDITA-2012-96
Evolution of Primordial Magnetic Fields from Phase Transitions
Tina Kahniashvili,1, 2, 3, ∗ Alexander G. Tevzadze,4, † Axel Brandenburg,5, 6, ‡ and Andrii Neronov7, 8, §
1
McWilliams Center for Cosmology and Department of Physics,
Carnegie Mellon University, 5000 Forbes Ave, Pittsburgh, PA 15213, USA
2
Department of Physics, Laurentian University, Ramsey Lake Road, Sudbury, ON P3E 2C,Canada
3
Abastumani Astrophysical Observatory, Ilia State University, 3-5 Cholokashvili Ave., Tbilisi, 0160, Georgia
4
Faculty of Exact and Natural Sciences, Tbilisi State University, 1 Chavchavadze Ave., Tbilisi, 0128, Georgia
5
Nordita, KTH Royal Institute of Technology and Stockholm University, Roslagstullsbacken 23, 10691 Stockholm, Sweden
6
Department of Astronomy, Stockholm University, 10691 Stockholm, Sweden
7
ISDC Data Centre for Astrophysics, Ch. d’Ecogia 16, 1290 Versoix, Switzerland
8
Geneva Observatory, Ch. des Maillettes 51, 1290 Sauverny, Switzerland
(MARCH 12, 2013, Revision: 1.47 )
We consider the evolution of primordial magnetic fields generated during cosmological, electroweak
or QCD, phase transitions. We assume that the magnetic field generation can be described as an
injection of magnetic energy to cosmological plasma at a given scale determined by the moment
of magnetic field generation. A high Reynolds number ensures strong coupling between magnetic
field and fluid motions. The subsequent evolution of the magnetic field is governed by decaying
hydromagnetic turbulence. Both our numerical simulations and a phenomenological description
allow us to recover “universal” laws for the decay of magnetic energy and the growth of magnetic
correlation length in the turbulent (low viscosity) regime. In particular, we show that during the
radiation dominated epoch, energy and correlation length of non-helical magnetic fields scale as
conformal time to the powers −1/2 and +1/2, respectively. For helical magnetic fields, energy and
correlation length scale as conformal time to the powers −1/3 and +2/3, respectively. The universal
decay law of the magnetic field implies that the strength of magnetic field generated during the QCD
phase transition could reach ∼ 10−9 G with the present day correlation length ∼ 50 kpc. The fields
generated at the electroweak phase transition could be as strong as ∼ 10−10 G with correlation
lengths reaching ∼ 0.3 kpc. These values of the magnetic fields are consistent with the lower bounds
of the extragalactic magnetic fields.
PACS numbers: 98.70.Vc, 98.80.-k
I.
INTRODUCTION
Astronomical observations show that galaxies have
magnetic fields with a component that is coherent over
a large fraction of the galaxy with field strengths of the
order of 10−6 G; see Refs. [1–3] and references therein.
Understanding the origin of these fields is one of the
challenging questions of modern astrophysics. Generally
speaking, there are two popular scenarios. The first one
envisages the generation of magnetic fields through different astrophysical mechanisms. More precisely, it is
assumed that an initially tiny magnetic field is produced
through a battery mechanism [4]. The correlation length
of such a field is limited by galactic length scales. The
second scenario to explain the origin of the magnetic field
in galaxies and clusters presumes that the observed magnetic fields were amplified from cosmological weak seed
magnetic fields [5]. In this case the correlation length of
such a seed field might be as large as the horizon scale today if we admit that the field has been generated during
∗ Electronic
address:
address:
‡ Electronic address:
§ Electronic address:
† Electronic
[email protected]
[email protected]
[email protected]
[email protected]
inflation. There are different possibilities for seed magnetic field amplification ranging from a magnetohydrodynamic (MHD) dynamo to the adiabatic compression
of magnetic field lines during structure formation [6, 7].
Galactic magnetic fields are usually measured through
the induced Faraday rotation effect [2, 3] and, as mentioned above, the field magnitude is of the order of a few
10−6 G at typical scales of 10 kpc. The primordial magnetic energy density contributes to the radiation field,
and thus the big bang nucleosynthesis (BBN) bound implies ΩB h20 ≤ 2.4 × 10−6 [8] if the magnetic field has been
generated prior to BBN. If the correlation length of the
magnetic field is much larger than λB & 1 Mpc, smaller
limits on the magnetic field energy density arise from the
cosmological data, making Bmax ≤ a few 10−9 G; see [9]
and references therein. A correlation length-dependent
lower limit on magnetic fields in the intergalactic medium
(IGM) could be derived from gamma-ray observations of
blazars [10–13]. In the limit of large correlation lengths,
λB & 1 Mpc, the bound is at the level of 10−17 G (see
also Ref. [14] for a discussions on possible uncertainties
in the measurements of blazar spectra).
It is possible, in principle, that the IGM magnetic
fields are of primordial origin. Another possibility is that
the fields are spread through the IGM by outflows from
galaxies at late stages of the evolution of the Universe.
To distinguish between these two possibilities, it is im-
2
portant to identify measurable characteristics of the IGM
magnetic fields which are different in the two cases and
to study the possibility of measuring such characteristics.
In this paper we consider the observational properties
of IGM magnetic fields expected if the fields originate
from cosmological phase transitions (PT) such as the
electroweak (EW) and QCD PTs [15, 17]. Some scenarios of magnetic field generation during EWPT or QCDPT
also produce magnetic helicity [16]. We follow the evolution of fields from the epoch right after the magnetogenesis up to the present day epoch. Our approach is different from that adopted in the previous studies, which
mostly concentrated on the analysis of the range of possible field strengths at a pre-defined scale of interest (e.g.
1 Mpc). Instead, we study the evolution of field characteristics that are most relevant for measurements using radio and gamma-ray astronomy. Specifically, we are
interested in the evolution of the magnetic energy density ρB , which
determines the characteristic field strength
√
B (eff) = 8πρB , and the characteristic correlation scale
(integral scale) λM at which the field reaches the strength
B (eff) .
The present day integral scale depends on the temperature T⋆ and the number of relativistic degrees of freedom
g⋆ at the moment when the primordial magnetic field is
generated. These parameters determine the maximal allowed value of the magnetic energy density injected in the
PT plasma, as well as the initial correlation length of the
magnetic field [18]. We do not separately consider the
effect of helicity transfer related to the chiral anomaly,
which might be important in the presence of strong magnetic fields at temperatures above 10–100 MeV [19]. Such
a transfer could be considered as part of the magnetogenesis process which could persist all the way down to
the temperature scale of the QCD phase transition. We
only use fundamental physical laws, such as conservation
of energy, and how the magnetic field interacts with the
cosmological plasma through MHD turbulence, and do
not make any assumption about the physical processes
responsible for the primordial magnetic field generation.
In Sec. II we give an overview of the spatial and temporal characteristics of the primordial magnetic field. The
results of our analysis are presented in Sec. III, where
we discuss the evolution of the magnetic field. Conclusions are presented in Sec. IV. We employ natural units
with ~ = 1 = c and gaussian units for electromagnetic
quantities.
II.
A.
MODEL DESCRIPTION
Effective Magnetic Field Characteristics
We assume that the phase transition-generated magnetic fields satisfy the causality condition [17, 22, 23].
The maximal correlation length ξmax for a causally generated primordial magnetic field cannot exceed the Hubble radius at the time of generation, H⋆−1 . Hence γ =
ξmax /H⋆−1 ≤ 1, where γ can be associated with the number of primordial magnetic field bubbles within the Hubble radius, N ∝ γ 3 . The comoving length (measured
today) corresponding to the Hubble radius at generation
is inversely proportional to the corresponding PT temperature T⋆ ,
1/6
100 GeV
100
λH⋆ = 5.8 × 10−10 Mpc
, (1)
T⋆
g⋆
and is equal to 0.5 pc for the QCDPT (with g⋆ = 15
and T⋆ = 0.15 GeV) and 6 × 10−4 pc for the EWPT
(with g⋆ = 100 and T⋆ = 100 GeV), and the comoving
primordial magnetic field correlation length ξmax ≤ λH .
This inequality assumes only the expansion of the Universe without accounting for MHD turbulence (free turbulence decay or an inverse cascade if a helical primordial
magnetic field is present). We also note that the number of PT bubbles within the Hubble radius is around 6
(γ ≃ 0.15) for QCDPT and around 100 (γ ≃ 0.01) for
EWPT. So the maximal correlation length ξmax is equal
to 0.08 pc for QCDPT and 6 × 10−6 pc for EWPT.
The maximal value of the primordial magnetic field energy density must satisfy the BBN bound, i.e. the total
energy density of the primordial magnetic field at nucleosynthesis ρB (where a should not exceed 10% of the
radiation energy density ρrad (aN ). In any case, the magnetic field is generated by mechanical motion of charged
particles, so that its energy density could constitute only
a fraction of matter energy density, which is at most comparable to the radiation energy density in the radiation
dominated Universe. Note that the maximal value of the
effective magnetic field is independent of the temperature
at generation T∗ , and depends only very weakly on the
number of relativistic degrees of freedom at the moment
of generation.
In what follows we are mostly interested in the evolution of the energy density of the magnetic field and
the length scale which gives the dominant contribution
to the energy density (the “integral scale”) during the
course of cosmological evolution. Taking this into account we adopt the following idealizing approximation.
We generate an initial primordial magnetic field by solving the MHD equations for a certain time during which
an external electromagnetic force is applied that is proportional to a delta function that peaks at the characteristic scale k0 = 2π/ξ0−1 . This corresponds to a magnetic
field with correlation length ξ0 . In this approximation
the characteristic
magnetic field strength at the scale ξ0
√
is B (eff) = 8πρB . We justify our assumption that ξ0
should be identified with the size of the largest magnetic
eddies by noting that the primordial magnetic field is
involved in MHD processes driven by turbulence. It is
natural to assume that the typical length scale of the
magnetic field generated during the PTs is determined
by the PT bubble size.
The dynamical evolution of the coupled magnetic field–
matter system leads to a spread of magnetic field over
a range of scales. The resulting power spectrum of
3
the magnetic field at small wavenumbers (or, equivalently, large distance scales) has the form of a power-law
PM (k) = EM(k)/(4πk 2 ) = Ak nB with a normalization
constant A and a slope1 nB . In particular, a white noise
power spectrum corresponds to nB = 0 [22], while the
Batchelor spectrum corresponds to nB = 2 [23]. The
power law spectrum extends up to a time-dependent integral scale ξM above which the power contained in the
magnetic field decreases rapidly due to turbulent decay
and/or viscosity damping.
Several previous studies, see [26] and references
therein, describe the primordial magnetic field in terms
of a smoothed (over length scale ξ) magnetic field Bξ
with Bξ2 = hBi (x)Bi (x)i|ξ . Knowing B (eff) and the slope
of the power spectrum, one can calculate the strength
of the smoothed magnetic field at any scale of interest ξ
[18]:
p
B (eff) Γ(nB /2 + 5/2)
Bξ =
.
(3)
(ξ/ξM )(nB +3)/2
The smoothed magnetic field might be of interest in the
context of certain problems. For example, the strength
of a magnetic field smoothed over a scale ξ ∼ 1 Mpc is
considered to be relevant in the context of seed magnetic
fields for galactic dynamos. It is, however, important to
note that Bξ is strongly nB dependent for a given value
of ξ. In particular, for causally-generated magnetic fields
with nB ≥ 0 there is significant “magnetic power” only
at small scales, and for ξ ≃ 1 Mpc the value of B1Mpc is
extremely small [27]. At the same time, it does not imply
that the magnetic field itself is weak. In fact, it could be
as strong as B (eff) ≃ 10−6 − 10−7 G, close to the bound
imposed by the BBN [18]. Only in the case of a scale
invariant magnetic field with nB → −3, generated for
example during inflation [28], Bξ is independent of ξ and
nB , and is equal to B (eff) . Note that sometimes (see, e.g.,
Eq. (8) in [21]], instead of calculating Bξ through B (eff) ,
the smoothed value of the magnetic field is determined
through the normalization constant A of the power spectrum as Bξ2 = AΓ(nB /2 + 3/2)/(ξM )(nB +3) /(2π)2 .
B.
Phenomenological description of the magnetic
field decay in the free turbulence regime
After generation, the evolution of the primordial magnetic field is a complex process affected by MHD as well
1
In general the magnetic field spectrum is determined through the
M (k) of the two-point correlation function of
Fourier transform Fij
the magnetic field, hBi (x)Bj (x + r)i, with the spectral function
[20]
M
Fij
(k) = Pij (k)
E M(k)
H M(k)
+ iεijl kl
.
2
4πk
8πk2
as by the expansion of the Universe [29–35, 37, 38]. In
our description, to account for the expansion of the Universe we make use of the fact that conformal invariance
allows for a description of MHD processes in the early
Universe by simply rescaling all physical quantities in
terms of their comoving values and using the conformal
time η [17]. After this procedure the MHD equations include the effects of the expansion while retaining their
conventional flat spacetime form.
The magnetic evolution process strongly depends on
initial conditions, as well as on the physical conditions of
the primordial plasma. We need to determine the scaling laws for the following magnetic field characteristics:
(i) magnetic energy density, (ii) correlation length, and
(iii) magnetic helicity. The magnetic energy and magnetic helicity spectra are related through the realizability condition, |HM(k, η)| ≤ 2EM
R (k, η)/k [31]. For the total magnetic
energy
E
(t)
=
EM(k, η) dk and helicity
M
R
HM (η) = HM(k, η) dk we get
HM (η) ≤ 2ξM (η)EM (η),
(4)
Z
(5)
where
ξM (η) ≡
−1
EM
(η)
EM (k, η)k −1 dk
is the comoving magnetic eddy correlation length (which
corresponds to the physical integral scale λM ), initially
set by the temperature at the magnetic field generation
moment ξM,in = λ0 = γλH⋆ ; see Eq. (1), and is independent of the presence of magnetic helicity.
Both helical and non-helical magnetic fields experience
large-scale MHD decay resulting in an increase of the
correlation length with a corresponding decrease in the
magnetic energy density at large scales; for a review see
[31, 39]. The time rate of this process depends strongly
on the presence of magnetic helicity [29, 30]. Taking this
into account, we consider the cases of helical and nonhelical magnetic fields separately.
As noted above, the initial magnetic field configuration
is given by a sharply peaked spectral energy density. The
coupling between primordial magnetic field and plasma,
which ensures the spreading of the fixed scale primordial magnetic field over a wide range of length scales,
forms a modified magnetic field spectrum within a few
turnover times (see Ref. [25] for more details). The final realization of the spectrum is given by the Batchelor spectrum, nB = 2. Our numerical simulation results are in perfect agreement with the “causality constraint” that in the cosmological context has been discussed in Ref. [17], and studied through analytical approach in Ref. [23]2 so that the power contained in the
large-scale modes is very small, while the total magnetic
(2)
Here Pij (k) = δij − ki kj /k2 , εijl is the antisymmetric tensor,
E M(k) and H M(k) are the magnetic energy and helicity spectra.
2
A recent study based on semi-analytical calculations [24] showed
the same shape for phase transition-generated magnetic fields.
In laboratory plasma as well as in numerical simulations, the
4
energy density is sufficiently large. MHD processes also
are responsible for the generation of fluid perturbations
when an initial magnetic field is present. These processes
finally result in equipartition between magnetic and kinetic energy densities [37, 40]. In contrast to the magnetic field, the velocity field has a white noise spectrum,
i.e. PK = EK /(4πk 2 ) = AK k nK with nK = 0 due to
the possible presence of longitudinal modes. To describe
adequately the evolution of fluid motions coupled to the
magnetic field we need to solve the complete set of MHD
equations; see below.
Below we present a phenomenological description of
the MHD decay laws at large scales. In the case of helical fields we mostly follow the description presented in
Secs. 4.2.3 and 5.3.2 of Ref. [31] and Sec. 7.3.4 of [41].
On scales below ξM , magnetic power is transferred to
smaller scales via the so-called direct cascade by turbulence until it is finally damped at the smallest scale ξd .
In MHD the magnetic field damping is usually determined through the Reynolds number as ξd /ξM = Re−3/4
[39]. The kinetic and magnetic Reynolds numbers in
the early Universe can be extremely high, and thus one
may expect that ξd ≪ ξM . In both helical and nonhelical cases the dissipative region of the energy density spectrum is given by a Kolmogorov-type spectrum3
2/3
EM = CK εM k −5/3 , where CK is a constant of order
unity (1.6–1.7 for a wide range of Reynolds numbers),
εM is the magnetic energy dissipation rate per unit mass,
Rk
given by εM = 2λ k0D k 2 EM (k) with λ being the magnetic diffusivity. At large scales above ξM the magnetic
field decay is strongly dependent on the presence of magnetic helicity. The high conductivity of plasma ensures
magnetic helicity conservation that is responsible for the
transfer of spectral energy from small to large scales via
a so-called inverse cascade. In the non-helical case the
process is more complicated. As we will see below, magnetic helicity conservations leads to a faster growth of the
correlation length and a slower decay of total magnetic
energy.
1.
Non-helical magnetic fields
As we already stated above, causality requires that
EM (k) ∝ k 4 and this is a consequence of the divergence-
free condition for the magnetic field [23]. On the other
hand, there is no zero-divergence requirement for the velocity field, and this allows for the possibility to have a
white noise spectrum for the velocity field, i.e. EK (k) ∝
k 2 ; see [22]. We would like to note that our numerical
simulations allowing the longitudinal forcing, see [37, 40],
show a white noise spectrum for the velocity field as a final configuration. Under these conditions, the power of
magnetic field modes on the large scales is much smaller
than the power of plasma motions. Thus, potentially the
magnetic field might be amplified via a transfer of energy
from plasma motions at large scales. The time scale η on
which the field can be amplified at large scales can be
deduced from the induction equation
∂B
vK (L)B
∼
→ η ∼ L/vL
∂t
L
i.e., the characteristic field amplification time scale is approximately the plasma eddy turnover time. On this time
scale an equipartition between kinetic and magnetic energies could be reached over the distance range L. In the
final configuration the equipartition between magnetic
and kinetic energies is a consequence of the coupling between magnetic and velocity fields.4 The growth of the
magnetic field up to equipartition with the fluid on large
scales is somewhat similar to the phenomenon of an “inverse cascade” of the magnetic power spectrum. Note
however that the source of power in the inverse transfer
of non-helical magnetic fields is different from the power
source in the case of helical fields. In the case of nonhelical fields, the power in the large wavelength modes
increases due to the presence of a large power reservoir
in the form of the turbulent motions of the plasma in
the same wavelength range. By contrast, in the case of
helical fields, the power on large scales grows due to the
transfer of power from the shorter wavelength modes.
Apart from the transfer of power from the plasma motions to the magnetic field at large scales, another effect
of evolution of plasma and magnetic field perturbations
is the turbulent decay of the power at short length scales
due to the phenomenon of the direct turbulent cascade.
At a given moment of time η, this phenomenon leads
to suppression of power in velocity and magnetic field
modes on scales smaller than the size of the largest processed eddy. The size of the largest processed eddy is
determined by the condition
ξK ∼ ξM ∼ vK (ξK )η,
3
resulting magnetic field spectrum at large scales can be given by a
white noise spectrum, nB = 0 [22] or even by a flatter Kazantsev
spectrum, nB = −1/2 [31]. However, here we consider the case
of a cosmological magnetic field for which the correlation length
is strongly limited by the Hubble horizon.
By a Kolmogorov-type spectrum we simply mean a k−5/3 spectrum and ignore anisotropies that are known to exist in nonhelical MHD [45]. Such a spectrum can be derived from a phenomenological approach too; see 5.3.2. of Ref. [31]. Our numerical simulations [25, 46] confirmed the Kolmogorov-type spectrum
for a wide range of magnetic Prandtl numbers
(6)
4
In fact, the two Reynolds numbers in
enough to ensure the validity of the
nomenological approach [32], according
similarity between kinetic and magnetic
tion (see Sec. 7.3.4 of Ref. [41]), i.e.
EM ∼ EK ∼ E,
(8)
the Universe are high
Kolmogorov-type pheto which there is selfenergy densities evolu(7)
5
where vL (ξ) is the characteristic velocity of the plasma
motions on the scale ξ. From the definition of the kinetic
energy power spectrum through the velocity two-point
correlation PK (k) ∼ |vk |2 ∼ |vK (L ∼ 1/k)|2 /k 2 one finds
that characteristic velocity vK on the distance scale L ∼
2
k −1 is vK
∼ k 2 PK . Since the power spectrum of plasma
perturbations is PK ∼ k 0 , we have vK ∼ k ∼ ξ −1 , so the
above equation has the solution
ξK ∼ ξM ∼ η 1/2 .
(9)
The energy of plasma perturbations on the scale ξK is
−2
EK = 4πk 2 PK ∼ k 2 ∼ ξK
∼ η −1 . Since the magnetic
field on the scale ξK ∼ ξM is in equipartition with the
plasma, the energy density of the magnetic field evolves
with time as
EM ∼ EK ∼ η −1 ,
so that the strength of magnetic field evolves as
p
B (eff) ∼ EM ∼ η −1/2 .
(10)
(11)
Below we demonstrate numerically (see Sec. III) that
the evolution laws ξM ∼ η 1/2 , EM ∼ η −1 are indeed realized in the free turbulence decay regime. It should be
noted that the “universality” of the decay law ξM ∝ η 1/2
is not realized if we were to consider the magnetic field
evolution separately from the velocity field evolution [31];
see also Refs. [34, 35] for the magnetic field decay laws
in the cosmological context. Accounting for the Loitsianskii invariant for turbulence leads to the decay laws being
dependent on the spectral shape; see Ref. [36] and references therein. It has also been claimed that the decay
laws in the case of non-helical magnetic fields strongly depend on the initial conditions and can be different even
when the helicity is extremely small [31].
2.
Helical fields
In the case of helical fields the evolution of EM and
ξM is determined directly byRthe condition of the conservation of magnetic helicity, A · B d3 x. High Reynolds
numbers allow us to follow the Kolmogorov-type phenomenological approach given above; see Sec. 4.2.3 of
Ref. [31]. Accounting for magnetic helicity conservation EM ξM = const5 and combining Eqs. (7) and ξM ∼
E 3/2 /ε, which follows from the dimensionless analysis
(based on the Kolmogorov-type approach), we get [32]
−
5
dEM
−5/2
∼ EM ,
dη
(12)
Accounting for magnetic helicity conservation and assuming that
the magnetic spectral energy is sharply peaked at ξM a very
−1/2
rough estimate implies that B eff ∝ ξM .
which leads to the decay laws EM ∝ η −2/3 and ξM ∝
η 2/3 . Below we demonstrate numerically (see Sec. III B)
the appearance of the ξM ∼ t2/3 , EM ∼ t−2/3 laws in
the evolution of helical magnetic fields in the free decay
regime.
C.
Simulations setup
To model the evolution of the magnetic field and fluid
perturbations we solve the compressible equations with
the pressure√given by p = ρc2s , where ρ is the gas density
and cs = 1/ 3 is the sound speed for an ultra-relativistic
gas. Following our earlier work [37], we solve the governing equations for the logarithmic density ln ρ, the velocity
v, and the magnetic vector potential A, in the form
D ln ρ
= −∇ · v,
Dη
Dv
= J × B − c2s ∇ ln ρ + fvisc ,
Dη
∂A
= v × B + fM + λ∇2 A,
∂η
(13)
(14)
(15)
where D/Dη = ∂/∂η + v · ∇ is the advective derivative,
fvisc = ν ∇2 v + 13 ∇∇ · v + G is the viscous force in
the compressible case with constant kinematic viscosity
ν and Gi = 2Sij ∇j ln ρ as well as Sij = 21 (vi,j + vj,i ) −
1
3 δij vk,k being the trace-free rate of strain tensor. Furthermore, J = ∇ × B/4π is the current density.
We use the Pencil Code [47] with a resolution of 5123
meshpoints. An important difference of our simulations
setup from those of previous studies is in the treatment
of the backreaction of fluid perturbations onto the magnetic field. Such treatment is important, especially on
large length scales, because the spectrum of velocity perturbations follows a white noise (EK ∝ k 2 ) spectrum
to large scales [22]. Large-scale fluid perturbations affect the magnetic field evolution at the largest scales and
could lead to a transfer of power to the large-scale modes
of the magnetic field, an effect similar to the inverse cascade developing even in the case of non-helical fields; see
also Sec. III.
Prior to the simulation of the magnetic field decay we
inject magnetic energy into the computational domain
at scales corresponding to the phase transition eddy size
see Ref. [37] for more details. We approximate the magnetic field injection by a delta function, allowing it to
interact through MHD processes with a rest plasma. After several turnover times the initial sharp peak of the
magnetic field starts to disappear and the magnetic field
begins to spread over a wide range of the wavenumbers;
see Fig. 4 of Ref. [37]. In a few turnover times the spectrum becomes established with a cut-off at small length
scales and a well defined Kolmogorov-like integral scale
kM , and a k 4 spectral shape at large length scale. In
contrast to the studies of Refs. [34, 35] we recover the
6
spectral shape of the magnetic field at large scale.6 This
is in perfect agreement with previous analytical results
of Ref. [23] based on causality and divergence free field
arguments.
We use simple power-law models for the decay of turbulence and scale the magnetic correlation length and
magnetic field with temperature as follows:
ξM
=
λ0
B (eff)
=
B⋆
T
T⋆
−nξ
T
T⋆
−nE
,
(16)
,
(17)
where B⋆ and T⋆ are the effective values of the magnetic
field at the moment of generation and the temperature
of the phase transition, respectively. Hence, the values of
the parameters nξ and nE describe the turbulent decay
laws that differ from each other in the non-helical and
helical cases.
Qualitative arguments presented above show that the
expected values for (nξ , nE ) for non-helical and helical
fields are (1/2, −1/2) and (2/3, −1/3), respectively. Below we show that this is indeed the case.
III.
SIMULATIONS RESULTS
In Sec. II we have presented a phenomenological description of the scaling laws for the magnetic correlation
length and the magnetic energy in the free turbulence
decay regime. Below we address the same scaling laws
based on our simulations. We also briefly review the results of previous works.
A.
Non-helical Magnetic Field Evolution
The scaling laws for the non-helical magnetic field evolution have been studied through different simulations by
different groups; see Refs. [29, 30, 32, 34, 48] and references therein. As is stated in Ref. [31], the magnetic
decay laws for the non-helical case strongly depend on
initial conditions, and result in exponents n in the decay
law EM (η) ∝ η −n that vary in the range 1.3 > n > 0.65.
Note that the numerical and phenomenological studies
performed in Refs. [34, 35] lead to ξM (η) ∝ η 0.4 and
EM (η) ∝ η −1.2 for a white noise spectrum, and this is
in good agreement with the grid turbulence description
of hydrodynamic turbulence [39]. On the other hand,
the 3D MHD simulations of Refs. [29, 37] and the phenomenological study of Ref. [33] show a slightly faster
6
Note that in Ref. [37] the spectral index between 3 and 4 due to
a different choice of the system parameters and run time.
FIG. 1: (color online). ξM (ξ) for helical (thin, red) and nonhelical (thick, blue) cases.
growth of the correlation length ξM (η) ∝ η 1/2 with a
magnetic energy decaying as EM ∝ t−1 . The difference
between two different scaling laws is probably due to different initial conditions. In particular, the initial velocity
field has traditionally been taken to be zero. By contrast, here we have taken as initial condition the result
of a self-consistent magnetically driven turbulence simulation. The numerical simulations of Ref. [46] show that
the growth of the correlation length is almost independent of the magnetic Prandtl number with ξM ∝ η 1/2
(Fig. 1), while the exponent of the total magnetic energy density decay is compatible with −1 (here closest
to −0.9; Fig. 2). Accounting for T ∝ a−1 and ∝ η −1
during the radiation dominated
epoch, and the magnetic
√
field strength B (eff) = 8πE M we get the scaling indices
for the decay of non-helical turbulence: nξ = 1/2 and
nE = −1/2. As expected, for k ≪ k0 early times, we
find EM (k, ξ) ∝ k 4 and EK (k, ξ) ∝ k 2 ; see Figure 3.
Furthermore, even in this non-helical case the spectral
energies increase with time for k ≪ k0 , while for k ≫ k0
they decrease.
B.
Helical Magnetic Field Evolution
As we have noted above, the presence of magnetic helicity results in the development of an inverse cascade
during which the correlation length is increasing while
the total magnetic energy decreases. Similar to the nonhelical case there are basically two different approaches:
(i) Refs. [32, 34, 35] assume exact conservation of magnetic helicity and the magnetic field is the dominant contribution to the total energy density, i.e., EK /EM ≪ 1
(where EK is the total kinetic energy density of turbulence); (ii) other approaches are given in Refs. [29, 33].
In particular, Ref. [29] refers to a more general case with
magnetic helicity evolving as HM (η) ∝ η −2s . Also, the
ratio between kinetic and magnetic energy densities has
in some studies assumed to be around 1; Ref. [33] assumes
7
FIG. 2: (color online). EM (ξ) (solid) and EK (ξ) (dashed) for
the helical (thin, red) and non-helical (thick, blue) cases.
FIG. 3: (color online). Evolution of EM (k, ξ) (solid) and
EK (k, ξ) (dashed) versus k for η = 5, 10, 20, 50, and 100
for the non-helical run. Thick lines are for η = 10. The red
4
dash-dotted lines give the k2 and
R k scalings for comparison.
All spectra are normalized by EM (k, 0) dk/k0 .
that the magnetic field evolves toward a force-free regime
with constant magnetic helicity and with a constant ratio
between magnetic and kinetic energy densities. All models [30, 32–35] show that the scaling laws are independent
of the initial magnetic field spectrum. In fact, there are
two main behaviors described: (i) Refs. [32, 34, 35] claim
EM (η) ∝ η −2/3 and ξM (η) ∝ η 2/3 ; (ii) Refs. [29, 33]
claim ξM (η) ∝ η 1/2 ; with EM (η) ∝ η −1/2 . In both scaling laws ξM EM ∼ const. The main difference between
these two scaling laws consists in choosing the turbulence
model. Refs. [29, 33] assume a force-free development
of the MHD turbulence decay, while Ref. [35], see their
Eq. (4), assumes a linear dependence between vorticity
and Lorentz force. Our new numerical results support
the former scenario (i). Fig. 3 shows the evolution of kinetic and magnetic spectral energies. As we can see, the
k 2 and k 4 laws are established at large scales for kinetic
and magnetic spectral energies, respectively. We can also
FIG. 4: (color online). Evolution of EM (k, ξ) (solid) and
EK (k, ξ) (dashed) versus k for η = 5, 10, 20, 50, and 100 for
the helical run. Thick lines are for η = 10. The red dash4
dotted lines give the k2 and
R k scalings for comparison. All
spectra are normalized by EM (k, 0) dk/k0 .
see the slight increase of power at large scales, even in the
case of non-helical fields.
We have performed a study of the large-scale decay
of a maximally helical magnetic field under conditions
similar to those in the non-helical case, and for different
magnetic Prandtl numbers as well as different values of
magnetic resistivity. We have recovered the EM (k) ∝ k 4
spectral shape and the scaling laws as ξM ∝ η 2/3 and
EM ∝ η −2/3 , so that nξ = 2/3, nE = −1/3 [49]; see also
[31] for more general discussion; see Figs. 1 and 2. Again
these scaling laws are valid when the correlation length
is greater than the damping scale, so that dissipation
does not play an important role. Similarly to the nonhelical run, we find for k ≪ k0 and early times that
EM (k, ξ) ∝ k 4 and EK (k, ξ) ∝ k 2 ; see Figure 4. In
this case there is a strong inverse cascade with a strong
increase of spectral energies with time for k ≪ k0 . We
can also see the constant magnetic power while the peak
is moving toward large scales (inverse cascade). This
corresponds to the constant helicity case.
IV.
IMPLICATION FOR COSMOLOGICAL
EVOLUTION OF MAGNETIC FIELD
Decay of cosmological MHD turbulence occurs together with the cooling of the Universe and the increase of
magnetic correlation length. Therewith, the correlation
length increases only up to the point when the Universe
reaches the temperature T = 1 eV [38]. The initial values
of correlation length and magnetic field strength, ξ0 and
B0 , at the temperature of magnetogenesis T∗ , together
with the two scaling indices nξ and nE , fully determine
the large-scale magnetic field decay, and as a result the
final configuration of the magnetic field.
Phenomenological arguments as well as numerical sim-
8
FIG. 5: (color online). The correlation length ξM (top
panel) and the maximal allowed Bmax (bottom panel) for
a primordial helical (thin red) and non-helical (thick blue)
magnetic fields generated during QCDPT. Constraints on
the magnetic field at T = 0.25eV are set to Bmax =
4.5 × 10−11 Gauss (ξM = 2 × 10−3 Mpc) and Bmax = 1.3 ×
10−9 Gauss (ξM = 5.6 × 10−2 Mpc) for non-helical and helical
cases, respectively. Dashed lines show areas where damping
processes may reduce ideal estimates.
FIG. 6: (color online). The correlation length ξM (top panel)
and the maximal allowed Bmax (bottom panel) for a primordial helical (thin red) and non-helical (thick blue) magnetic
fields generated during EWPT. Constraints on the magnetic
field at T = 0.25eV are set to Bmax = 1.3 × 10−12 Gauss
(ξM = 3.5 × 10−6 Mpc) and Bmax = 1.1 × 10−10 Gauss
(ξM = 3.1 × 10−4 Mpc) for non-helical and helical cases, respectively. Dashed lines show areas where damping processes
may reduce ideal estimates.
ulations show that nξ = 1/2 and nE = −1/2 in nonhelical case, while nξ = 2/3 and nE = −1/3 in the helical
case during the turbulent regime. The speed of growth of
ξM is constant, independently of the relation between ξM
and ξd . This implies that, in the case in which the initial
correlation length of magnetic field is comparable to the
size of cosmological horizon at the epoch of magnetogenesis, the final correlation length reaches 2×10−4 Mpc and
6 × 10−3 Mpc for non-helical magnetic fields generated
at EWPT and QCDPT, respectively. In the helical case,
the correlation length reaches 10 kpc for fields generated
at the QCDPT and 0.1 Mpc for the EWPT.
The evolutionary paths of magnetic field strength and
correlation length in time and in the B, ξM parameter
space are shown in Figs. 5–7. In principle, the free turbulence decay periods in the early Universe are intermittent, with periods of viscously damped evolution [34]. So,
starting from 10 MeV, Figs. 5 and 6 show only maximal
values of helical and non-helical magnetic fields and their
correlation length from QCD and EW phase transitions
without accounting for any possible damping processes
that can affect the given scaling laws. These maximal
values are derived using several assumptions: the magnetic correlation length during phase transition matches
the bubble size, and magnetic fields are excited with maximal amplitudes allowed by the BBN limit. In fact, before neutrino decoupling the viscous damping force fvisc
in Eq. (14) grows as ν∼ lmfp, ν ∼ η 4 , where lmfp, ν is the
neutrino mean free path. This growth is much faster
than the ξM ∼ η 1/2 growth of the integral scale of the
magnetic field. This means that, even if ξM ≫ lmfp, ν
at the moment of magnetogenesis, lmfp catches up with
ξM at a later time ηvisc . Starting from this time and
up to the moment of neutrino decoupling, the magnetic
field stops to decay, B ∼ const, because the fluid motions are damped by viscosity, v ∼ 0, so that there is
no coupling of magnetic field to the fluid in this regime.
However, turbulence re-starts after the neutrino decoupling, so that the system returns to the same evolutionary track shown in Fig. 7 just after neutrino decoupling.
At lower temperatures, when the viscosity is provided by
photon streaming, the viscous damping scale grows as
ν ∼ lmfp, γ ∼ η 2 , where lmfp, γ is the photon mean free
path. In this time interval the growth of ν is again faster
than the growth of ξM , so that the episode of viscously
damped evolution repeats when lmfp, γ reaches ξM . This
could again delay the advance of (B, ξB ) along the evolutionary track shown in Fig. 7. The end point of the
evolutionary track at the end of the radiation-dominated
era is well defined by the condition that the correlation
length of magnetic field should not be shorter than the
Silk damping scale times the Alfvén velocity [34]. The
loci of the possible end points of the evolution are shown
by the inclined thick solid (green/orange) lines in Fig. 7.
9
extremely low values of the smoothed magnetic field [27]
do not imply that the effective magnetic field in the range
1 pc–1 kpc are small enough to result in observational
changes in blazar emission spectra. The advantage of
using the effective magnetic field lies in its independence
of the spectral shape. In summary, if the magnetic field
has been generated during a phase transition, its correlation length is strongly limited. If future observations
were to detect a weak magnetic field ≤ 10−14 –10−15 G
with a typical correlation length of the order of a few
pc, this could serve as an indication of magnetogenesis
during EWPT, while a somewhat stronger field with a
correlation length of the order of kpc might indicate the
presence of QCDPT magnetogenesis.
Acknowledgments
FIG. 7: (color online). Cosmological evolution of B (eff) and
ξM for magnetic fields generated at the EWPT (green) and
QCDPT (orange). Arrows show the evolution of the strength
and integral scale of helical and non-helical fields during the
radiation dominated era up to their final values. Thick solid
line(s) show possible present day strength and integral scale
of the phase-transition generated magnetic fields.
V.
CONCLUSION
Our study shows that magnetic fields generated during
phase transitions are comparable with the observational
lower bound even if we account for large-scale decay as
well as additional Alfvén wave-induced damping. The
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